Yes, it is possible for you to apply to PhD programs in clinical psychology having done an MS in CMHC. Whether you will gain admission, which is what I think you are really asking, is hard to say. As you know, PhD programs in clinical psychology can be quite competitive, although just how competitive varies by program.
I'm in a counseling psych PhD program and half of the students we admit (we admit 7-8 per year) usually have a master's in psychology, counseling or something similar, sometimes they are licensed. PhD programs in clinical psychology may view the MS in CMHC differently though. I imagine it would depend on the program.
I don't think you will necessarily be any more or less competitive than you would coming straight from a BA/BS program, although this will obviously vary by program. The important elements would still apply: get plenty of research experience, clinical experience (which you will have gotten in the master's program), a well-written personal statement, outstanding letters of recommendation, high GPA and high test scores, etc.
You may also discover if you decide to pursue the master's that you are satisfied being a CMHC and decide not to go forward with the doctorate.